The Teensy is similar to the Arduino in that there are digital input/output pins and analog pins.
Connect the switches to 4 digital input pins. Which 4 doesn't really matter.
In loop(), read those 4 pins. If switch 1 is pressed, do something. If switch 2 is pressed, do something. Ditto for switches 3 and 4.
What, exactly does "send KEY_ENTER" mean? Do you want the Teensy to write those letters on a piece of paper, fold the paper to fit in an envelope, put the folded paper in the envelope, address it, stamp it, and deliver it to the post office? If so, you just might need more hardware.
The problem is not on the Arduino side; it's on the computer side. You need to somehow generate a keyboard event on the computer, based on things the Ardy is doing. There are two ways of doing this.
You can make certain boards (Arduino Uno, Mega) look like a keyboard on the USB bus by re-programming their USB chip. You will need a bit bashing programmer (like the AVR-ISP or STK500 or whatever) both to program the USB chip, and to program the Arduino after your serial USB function goes away. Load the keyboard firmware into the USB chip, then load the sketch into the Arduino, then plug it in. Also, debugging will be harder, because you won't have a serial port to look at. I don't know if the Teeny or Tiny support this function, though, as they may not have the appropriate USB interface chip.
Use a serial monitor program on the computer. This means that you first program the Arduino using the IDE. Then you quit the IDE, and start the serial monitor program. The monitor program opens the serial port, and then reads characters from the Ardy. When it sees the appropriate character, it posts the corresponding character on the system keyboard input bus. On Windows, you do this with SendInput() link: SendInput function (winuser.h) - Win32 apps | Microsoft Learn
The good news is that you can do this stepwise. First, make a sketch that sends the appropriate characters (say, A/B/C/D) for each button down and each button up (say, a/b/c/d). You can send this with Serial.print(). Use the built-in serial console to monitor this. Once that is working, your "arduino" bit is done, and you can start working on the compute bit.
At that point, write a computer program that opens a serial port, and posts keydown/keyup events based on receiving the right characters. You can test this by connecting to the serial port using some virtual serial loopback interface, for example (how to do this specifically differs by computer hardware and OS). That will allow you to debug the "keyboard sending" separate from the "arduino."
Once that is all debugged, plug the two together, and it Should Just Work (tm)!